Intereting Posts

Dijkstra's algorithm proof
Square root confusion: Why am I getting an answer if it doesn't work?
Orthogonal polynomials and Gram Schmidt
Largest idempotent
Prove $g(x+h) = g(x) + hg'(x) + \frac{1}{2} h^2 g''(x) + o(h^2)$ from definition of limit
$QR$ decomposition of rectangular block matrix
About the second fundamental form
Partial limits of sequence
About the Application of Cauchy-Schwarz (Basic): maximum of $3x+4y$ for $x^2+y^2 \leq 16$
Integration $I_n=\int _0^{\pi }\sin^{2n}\theta d\theta $
Number of occurrences of k consecutive 1's in a binary string of length n (containing only 1's and 0's)
Constructions of the smallest nonabelian group of odd order
The convergence of $\sqrt {2+\sqrt {2+\sqrt {2+\ldots}}}$
If a prime $p\mid ab$, then $p\mid a$ or $p\mid b$
Why are the limits of the integral $0$ and $s$, and not $-\infty$ and $+\infty$??

Use the Banach fixed point theorem to show that

the following sequence converges. What is the limit of this

sequence?

$$\left(\frac{1}{3},

\frac{1}{3+\frac{1}{3}},

\frac{1}{3+\frac{1}{3+\frac{1}{3}}},

\dots\right)$$

I inferred the closed form of the sequence

$$\begin{align}

x_0 &= \frac{1}{3} \\

x_n &= \frac{1}{3 + x_{n-1}}

\end{align}$$

and then the function that would, I assume, be that which would be applied repeatedly to converge the sequence (per the Banach fixed point theorem)

$$f(x)=\frac{1}{3+x}$$

- Proof: A convergent Sequence is bounded
- Proof by induction that $\sum\limits_{k=1}^n \frac{1}{3^k}$ converges to $\frac{1}{2}$
- If $(a_n)$ is such that $\sum_{n=1}^\infty a_nb_n$ converges for every $b\in\ell_2$, then $a\in\ell_2$
- Boundedness and pointwise convergence imply weak convergence in $\ell^p$
- A sequence converges if and only if every subsequence converges?
- Convergence/Divergence of some series

That’s about as far as I know to go *for sure*, but I assumed then that I needed to prove $f(x)$ was a contraction mapping (as the hypothesis of the Banach fixed point theorem dictates), that is

$$d(f(x),f(y)) \leq r d(x,y)$$

where $r \in [0, 1)$, and so

$$ \begin{align}

\frac{1}{3+x} – \frac{1}{3+y} &\leq r(x – y) \\

\frac{1}{(x – y)(3 + x)} – \frac{1}{(x – y)(3 + y)} &\leq r \\

\frac{-1}{(x + 3)(y + 3)} &\leq r

\end{align}$$

Although I don’t think the above proves anything useful at all. (Unless $(x+3)(y+3) < -1$, where it is shown that $\exists r \in [0,1) : \dots$.)

- Sequence of convex functions
- If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \frac{n-1}{n+1}$, find $a_n$ and the sum.
- Convergence of Series
- does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?
- Methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$
- Covergence test of $\sum_{n\geq 1}{\frac{|\sin n|}{n}}$
- Why *all* $\epsilon > 0$, in the $\varepsilon-\delta$ limit definition?
- $a_n$ is bounded and decreasing
- How to find the partial sum of a given series?
- Recursive sequence convergence with trigonometric terms

You’re on the right track (though you forgot the rather crucial absolute value signs in your inequality). Try showing that $f$ is a contraction map on some *subinterval* of $\Bbb R$. (It isn’t even defined at $-3$, so we have to consider a proper subinterval, anyway.) Since we’re taking our initial point to be $\frac13$, we might as well consider $[0,\infty)$. We could instead use $[\alpha,\infty)$ for some $-3<\alpha<0$, if we liked, but $0$ works just fine.

Show that if $x\in[0,\infty)$ then $f(x)\in[0,\infty).$ Then for all $x,y\in[0,\infty)$, you can see that $$\left|\frac1{3+x}-\frac1{3+y}\right|=\left|\frac{y-x}{(3+x)(3+y)}\right|=\frac1{(3+x)(3+y)}|x-y|\le r|x-y|,$$ where $r=???$

**Hint:** Use the mean value theorem to prove your map $f(x)=\frac{1}{3+x}$ is contraction on the interval $[0,\infty)$. See here for detailed techniques.

- Two different solutions to integral
- Can you derive a formula for the semiprime counting function from the prime number theorem?
- A 10-digit number whose $n$th digit gives the number of $(n-1)$s in it
- Is there any perfect squares that are also binomial coefficients?
- An entire function which is real on the real axis and map upper half plane to upper half plane
- Is this determinant identity true?
- What the implicit function theorem is actually showing
- Is such a field element an element of a subring?
- Solovay's Model and Choice
- Find $\min(\operatorname{trace}(AA^T))$ for invertible $A_{n\times n}$
- How to find the maximum diagonal length inside a dodecahedron?
- Finding the equation of the normal line
- Let $4$ and $5$ be the only eigenvalues of $T$. Show $T^2-9T + 20I = 0$ , T is self adjoint.
- Probability of picking a random natural number
- A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$